---
tag: 0R47MAD57S
kind: definition
name: Monoid
msc: [20M10]
defines: 0N6R902FG1
uses: [0V3CFG1KK9, 0RQPZZ44T4]
sources:
  - url: https://en.wikipedia.org/wiki/Monoid
    locator: Definition
---

## Definition

A monoid is a triple $(M, \ast, e)$, where $(M, \ast)$ is a [[semigroup|0XMEM406JM]] and $e \in M$ is an [[identity element|07WYQPB8E3]] for $\ast$.
Thus $(a \ast b) \ast c = a \ast (b \ast c)$ and $e \ast a = a \ast e = a$ for all $a, b, c \in M$.

## Remarks

The natural numbers $\mathbb{N} = \{0, 1, 2, \ldots\}$ with addition form a monoid with identity $0$.
The integers with multiplication form a monoid with identity $1$.
The identity element of a monoid is unique, thus some authors write a monoid as the pair $(M, \ast)$.
Each [[group|G5S1M9]] is a monoid in which each element has an [[inverse|0ZNZH0EXFX]].

## Explanations

### L1 human

A monoid is a set with an associative operation and an element that does nothing.
The strings of letters with concatenation form a monoid, and the empty string is the identity.
In a monoid, an element need not have an inverse: you cannot undo the addition of a letter.